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Faster Algorithms for BoundConsistency of the Sortedness and the Alldifferent Constraint
 In Proceedings of the Sixth International Conference on Principles and Practice of Constraint Programming
, 2000
"... We present narrowing algorithms for the sortedness and the alldifferent constraint which achieve boundconsistency. The algorithm for the sortedness constraint takes as input 2n intervals X1 , ..., Xn , Y1 , ..., Yn from a linearly ordered set D. Let S denote the set of all tuples t 2 X1 Xn Y1 ..."
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Cited by 35 (2 self)
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We present narrowing algorithms for the sortedness and the alldifferent constraint which achieve boundconsistency. The algorithm for the sortedness constraint takes as input 2n intervals X1 , ..., Xn , Y1 , ..., Yn from a linearly ordered set D. Let S denote the set of all tuples t 2 X1 Xn Y1 Yn such that the last n components of t are obtained by sorting the first n components. Our algorithm determines whether S is nonempty and if so reduces the intervals to boundconsistency. The running time of the algorithm is asymptotically the same as for sorting the interval endpoints. In problems where this is faster than O(n log n), this improves upon previous results. The algorithm for the alldifferent constraint takes as input n integer intervals Z1 , ..., Zn . Let T denote all tuples t 2 Z1 Zn where all components are pairwise different. The algorithm checks whether T is nonempty and if so reduces the ranges to boundconsistency. The running time is also asymptotically the same as for sorting the interval endpoints. When the constraint is for example a permutation constraint, i.e. Z i [1; n] for all i, the running time is linear. This also improves upon previous results.
Augment or Push? A computational study of Bipartite Matching and Unit Capacity Flow Algorithms
 ACM J. EXP. ALGORITHMICS
, 1998
"... We conduct a computational study of unit capacity flow and bipartite matching algorithms. Our goal is to determine which variant of the pushrelabel method is most efficient in practice and to compare pushrelabel algorithms with augmenting path algorithms. We have implemented and compared three pus ..."
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Cited by 31 (1 self)
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We conduct a computational study of unit capacity flow and bipartite matching algorithms. Our goal is to determine which variant of the pushrelabel method is most efficient in practice and to compare pushrelabel algorithms with augmenting path algorithms. We have implemented and compared three pushrelabel algorithms, three augmenting path algorithms (one of which is new), and one augmentrelabel algorithm. The depthfirst search augmenting path algorithm was thought to be a good choice for the bipartite matching problem, but our study shows that it is not robust. For the problems we study, our implementations of the fifo and lowestlevel selection pushrelabel algorithms have the most robust asymptotic rate of growth and work best overall. Augmenting path algorithms, although not as robust, on some problem classes are faster by a moderate constant factor. Our study includes several new problem families and input graphs with as many as 5 \Theta 10 5 vertices.
A new combinatorial approach to sparse graph problems
 In Proc. ICALP
, 2008
"... Abstract. We give a new combinatorial data structure for representing arbitrary Boolean matrices. After a short preprocessing phase, the data structure can perform fast vector multiplications with a given matrix, where the runtime depends on the sparsity of the input vector. The data structure can a ..."
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Abstract. We give a new combinatorial data structure for representing arbitrary Boolean matrices. After a short preprocessing phase, the data structure can perform fast vector multiplications with a given matrix, where the runtime depends on the sparsity of the input vector. The data structure can also return minimum witnesses for the matrixvector product. Our approach is simple and implementable: the data structure works by precomputing small problems and recombining them in a novel way. It can be easily plugged into existing algorithms, achieving an asymptotic speedup over previous results. As a consequence, we achieve new running time bounds for computing the transitive closure of a graph, all pairs shortest paths on unweighted undirected graphs, and finding a maximum nodeweighted triangle. Furthermore, any asymptotic improvement on our algorithms would imply a o(n 3 / log 2 n) combinatorial algorithm for Boolean matrix multiplication, a longstanding open problem in the area. We also use the data structure to give the first asymptotic improvement over O(mn) for all pairs least common ancestors on directed acyclic graphs. 1
An n log n algorithm for hyperminimizing a (minimized) deterministic automaton
 THEOR. COMPUT. SCI
, 2010
"... We improve a recent result [Badr: Hyperminimization in O(n²). Int. J. Found. Comput. Sci. 20, 2009] for hyperminimized finite automata. Namely, we present an O(n log n) algorithm that computes for a given deterministic finite automaton (dfa) an almostequivalent dfa that is as small as possible suc ..."
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Cited by 7 (4 self)
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We improve a recent result [Badr: Hyperminimization in O(n²). Int. J. Found. Comput. Sci. 20, 2009] for hyperminimized finite automata. Namely, we present an O(n log n) algorithm that computes for a given deterministic finite automaton (dfa) an almostequivalent dfa that is as small as possible such an automaton is called hyperminimal. Here two finite automata are almostequivalent if and only if the symmetric difference of their languages is finite. In other words, two almostequivalent automata disagree on acceptance on finitely many inputs. In this way, we solve an open problem stated in [Badr, Geffert, Shipman: Hyperminimizing minimized deterministic finite state automata. RAIRO Theor. Inf. Appl. 43, 2009] and by Badr. Moreover, we show that minimization linearly reduces to hyperminimization, which shows that the timebound O(n log n) is optimal for hyperminimization. Independently, similar results were obtained in [Gawrychowski, Jez: Hyperminimisation made efficient. Proc. MFCS,
Matching Algorithms Are Fast in Sparse Random Graphs
 THEORY OF COMPUTING SYSTEMS
, 2005
"... We present an improved average case analysis of the maximum cardinality matching problem. We show that in a bipartite or general random graph on n vertices, with high probability every nonmaximum matching has an augmenting path of length O(log n). This implies that augmenting path algorithms like t ..."
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We present an improved average case analysis of the maximum cardinality matching problem. We show that in a bipartite or general random graph on n vertices, with high probability every nonmaximum matching has an augmenting path of length O(log n). This implies that augmenting path algorithms like the Hopcroft–Karp algorithm for bipartite graphs and the Micali–Vazirani algorithm for general graphs, which have a worst case running time of O(m √ n), run in time O(m log n) with high probability, where m is the number of edges in the graph. Motwani proved these results for random graphs when the average degree is at least ln(n)
A scaling algorithm for maximum weight matching in bipartite graphs
 IN: PROCEEDINGS 23RD ACMSIAM SYMPOSIUM ON DISCRETE ALGORITHMS (SODA
"... Given a weighted bipartite graph, the maximum weight matching (MWM) problem is to find a set of vertexdisjoint edges with maximum weight. We present a new scaling algorithm that runs in O(m √ n log N) time, when the weights are integers within the range of [0, N]. The result improves the previous b ..."
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Cited by 6 (1 self)
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Given a weighted bipartite graph, the maximum weight matching (MWM) problem is to find a set of vertexdisjoint edges with maximum weight. We present a new scaling algorithm that runs in O(m √ n log N) time, when the weights are integers within the range of [0, N]. The result improves the previous bounds of O(Nm √ n) by Gabow and O(m √ n log (nN)) by Gabow and Tarjan over 20 years ago. Our improvement draws ideas from a not widely known result, the primal method by Balinski and Gomory.
Maximum flows by incremental breadthfirst search
 IN ESA, LNCS 6942
, 2011
"... Maximum flow and minimum st cut algorithms are used to solve several fundamental problems in computer vision. These problems have special structure, and standard techniques perform worse than the specialpurpose BoykovKolmogorov (BK) algorithm. We introduce the incremental breadthfirst search (I ..."
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Cited by 6 (1 self)
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Maximum flow and minimum st cut algorithms are used to solve several fundamental problems in computer vision. These problems have special structure, and standard techniques perform worse than the specialpurpose BoykovKolmogorov (BK) algorithm. We introduce the incremental breadthfirst search (IBFS) method, which uses ideas from BK but augments on shortest paths. IBFS is theoretically justified (runs in polynomial time) and usually outperforms BK on vision problems.
Scaling algorithms for approximate and exact maximum weight matching
, 2011
"... The maximum cardinality and maximum weight matching problems can be solved in time Õ(m √ n), a bound that has resisted improvement despite decades of research. (Here m and n are the number of edges and vertices.) In this article we demonstrate that this “m √ n barrier ” is extremely fragile, in the ..."
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Cited by 3 (0 self)
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The maximum cardinality and maximum weight matching problems can be solved in time Õ(m √ n), a bound that has resisted improvement despite decades of research. (Here m and n are the number of edges and vertices.) In this article we demonstrate that this “m √ n barrier ” is extremely fragile, in the following sense. For any ɛ> 0, we give an algorithm that computes a (1 − ɛ)approximate maximum weight matching in O(mɛ −1 log ɛ −1) time, that is, optimal linear time for any fixed ɛ. Our algorithm is dramatically simpler than the best exact maximum weight matching algorithms on general graphs and should be appealing in all applications that can tolerate a negligible relative error. Our second contribution is a new exact maximum weight matching algorithm for integerweighted bipartite graphs that runs in time O(m √ n log N). This improves on the O(Nm √ n)time and O(m √ n log(nN))time algorithms known since the mid 1980s, for 1 ≪ log N ≪ log n. Here N is the maximum integer edge weight. 1
Efficient Algorithms for Path Problems in Weighted Graphs
, 2008
"... Problems related to computing optimal paths have been abundant in computer science since its emergence as a field. Yet for a large number of such problems we still do not know whether the stateoftheart algorithms are the best possible. A notable example of this phenomenon is the all pairs shorte ..."
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Cited by 3 (0 self)
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Problems related to computing optimal paths have been abundant in computer science since its emergence as a field. Yet for a large number of such problems we still do not know whether the stateoftheart algorithms are the best possible. A notable example of this phenomenon is the all pairs shortest paths problem in a directed graph with real edge weights. The best algorithm (modulo small polylogarithmic improvements) for this problem runs in cubic time, a running time known since the 1960s (by Floyd and Warshall). Our grasp of many such fundamental algorithmic questions is far from optimal, and the major goal of this thesis is to bring some new insights into efficiently solving path problems in graphs. We focus on several path problems optimizing different measures: shortest paths, maximum bottleneck paths, minimum nondecreasing paths, and various extensions. For the allpairs versions of these path problems we use an algebraic approach. We obtain improved algorithms using reductions
Incremental Topological Ordering and Strong Component Maintenance
, 2008
"... Abstract. We present an online algorithm for maintaining a topological order of a directed acyclic graph as arcs are added, and detecting a cycle when one is created. Our algorithm takes O(m 1/2) amortized time per arc, where m is the total number of arcs. For sparse graphs, this bound improves the ..."
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Abstract. We present an online algorithm for maintaining a topological order of a directed acyclic graph as arcs are added, and detecting a cycle when one is created. Our algorithm takes O(m 1/2) amortized time per arc, where m is the total number of arcs. For sparse graphs, this bound improves the best previous bound by a logarithmic factor and is tight to within a constant factor for a natural class of algorithms that includes all the existing ones. Our main insight is that the bidirectional search method of previous algorithms does not require an ordered search, but can be more general. This allows us to avoid the use of heaps (priority queues) entirely. Instead, the deterministic version of our algorithm uses (approximate) medianfinding. The randomized version of our algorithm avoids this complication, making it very simple. We extend our topological ordering algorithm to give the first detailed algorithm for maintaining the strong components of a directed graph, and a topological order of these components, as arcs are added. This extension also has an amortized time bound of O(m 1/2) per arc. 1